The Upper And Lower Bound Estimations Of Nonlinear Optimization Problems

For the bilevel nonlinear problems, most of the solving methods are based on the special structure of the problem. This paper presents two classes of different problems that were solved with different bilevel programming methods. First of all, for uncertain parameters Generalized Geometric Programming, since the parameters of the objective function and constraint function are in some intervals, it is difficulty to obtain the optimizing objective function value, therefore, the best way is to find the upper and lower bounds of the objective function. To solve the problem, this paper presents a generalized bilevel programming approach. First of all, the upper and lower bounds of the original problem can be expressed in generalized bilevel programming. Then can be transformed into equivalent monotonic optimization problems with the characteristics of the problem itself. Finally, the traditional branch-reduce-and bound method can obtained the optimal solution and the optimal value of the problem. To the second problem that is an interval quadratic programming, quadratic matrix Q is the variable, and each element is in an interval, rather than constant, other parameters (cost coefficients, constraint coefficients and right hand side) are also interval. Therefore, the objective function is also an interval number; we need to get the upper and lower bounds of objective function value by two bilevel mathematical programmings. According to the properties of the problem itself, the lower bound of the objective function value can be solved. The upper bound can expressed as the difference between two positive definite matrices. Using the linear estimate function for the concave part, the original problem converts into a convex programming problem. To solve this problem, the common method is dual programming. This method can not only solve quadratic programming problem effectively, but also makes the solving process easily: Finally, a simple example shows the feasibility and effectiveness of the proposed algorithm.